## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 2, Number 3 (1992), 519-535.

### Poisson Process Approximations for the Ewens Sampling Formula

Richard Arratia, A. D. Barbour, and Simon Tavare

#### Abstract

The Ewens sampling formula is a family of measures on permutations, that arises in population genetics, Bayesian statistics and many other applications. This family is indexed by a parameter $\theta > 0$; the usual uniform measure is included as the special case $\theta = 1$. Under the Ewens sampling formula with parameter $\theta$, the process of cycle counts $(C_1(n), C_2(n), \ldots, C_n(n), 0, 0, \ldots)$ converges to a Poisson process $(Z_1, Z_2, \ldots)$ with independent coordinates and $\mathbb{E}Z_j = \theta/j$. Exploiting a particular coupling, we give simple explicit upper bounds for the Wasserstein and total variation distances between the laws of $(C_1(n), \ldots, C_b(n))$ and $(Z_1, \ldots, Z_b)$. This Poisson approximation can be used to give simple proofs of limit theorems with bounds for a wide variety of functionals of such random permutations.

#### Article information

**Source**

Ann. Appl. Probab., Volume 2, Number 3 (1992), 519-535.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005647

**Digital Object Identifier**

doi:10.1214/aoap/1177005647

**Mathematical Reviews number (MathSciNet)**

MR1177897

**Zentralblatt MATH identifier**

0756.60006

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 05A05: Permutations, words, matrices 05A16: Asymptotic enumeration 92D10: Genetics {For genetic algebras, see 17D92}

**Keywords**

Total variation population genetics permutations

#### Citation

Arratia, Richard; Barbour, A. D.; Tavare, Simon. Poisson Process Approximations for the Ewens Sampling Formula. Ann. Appl. Probab. 2 (1992), no. 3, 519--535. doi:10.1214/aoap/1177005647. https://projecteuclid.org/euclid.aoap/1177005647